Finite Element Analysis for Singularly Perturbed Advection-Diffusion Robin Boundary Values Problem ()
1. Introduction
We consider the singularly perturbed advection-diffusion Robin boundary values problem
(1)
(2)
with sufficiently smooth functions
, and a small positive parameter
. We assume that 
be decreasing monotonously, moreover
(3)
which guarantees the unique solvability of the problem. It is well known that there exists a boundary layer of width 
at 
(see [1], K.W. Chang & F.A. Howes 1984). Standard numerical methods for singularly perturbed problem exhibit spurious error unless the layeradapted-mesh, such as Shishkin mesh, B-mesh(see [2-7]) are employed, for the solutions of singularly perturbed problem usually contain layers. The main objective of the paper is to use the method of singular perturbation to give the estimation of error between solution and the finite element approximation w.r.t. some energy norm on shishkin-type mesh.
Throughout the paper, we shall use C to denote a generic positive constant ,that is independent of ε and mesh, while it can value differently at different places, we occasionally use a subscribed one such as C1.
2. Properties of Solution for Continuous Problem
In this section, some properties and bounds of the exact solution and its derivatives are deduced preliminarily.
Lemma 1 (Maximum principle) Let 
If
for
, 
,then
for 
Proof. Assume that there exists 
such that
If
, then there holds 
which results in a contradiction to
;Thus
.
Since we have 
the differential operator on 
at 
gives
which result in a contradiction to 
therefore we can conclude that the minimum of 
is non-negative.
Lemma 2 (Comparison principle) If 
satisfy 
for
, and
,
, then 
for all
.
Lemma 3 (Stability result) If
, then we have
for all
.
The Proofs of Lemma 2 and Lemma 3 are followed essentially from Lemma 1. (See [3] Roos, Stynes and Tobiska, (1996)).
Lemma 4 Let 
be the solution to (1) (2). then there exists a constant C, such that for all
, we have the splitting
(4)
where the regular component u(x) satisfy
(5)
while the layer component 
satisfy
. (6)
Proof. It is known that (see [4] Kellogg 1978, Chang & Howes 1984)
We assume 
spontaneously since singular perturbation.
We set 
such that 
and 
on 
thus 
on
and then extended on (0,1) with
;
Next let
Then considering that 
on
, we know that 
satisfy
on 
3. Simplification
For simplification of the original problem, we set a transformation
then Equation (1), (2) are transformed to
Continuing, we transform the boundary values homogeneously by
at last, the problem (1), (2) are converted to
where in the 
posses the same properties as
, thus we just make discussion on the simplified problem below
(1’)
(2’)
4. The Analysis of Finite Element Approximation
We consider the Galerkin approximation in form of Find 
such that
(7)
where
, the bilinear form
And a natural norm associated with 
is chosen by
wherein
is the usual 2-norm.
It is easy to see that 
is coercive with respect to 
by the assumption of the monotony of 
which guarantees the existence of the solution of (7) (see [8-10]). Let N be an even positive integer that denotes the number of mesh intervals.
We consider the space of piecewise linear function denoted by 
as our work space, 
denotes the piecewise linear interpolant to 
at some special mesh points on I, We’ll utmost estimate the error
.
Firstly we have
(8)
For the second term of inequality (8), we make use of the coerciveness, continuousness of 
and the Galerkin orthogonality relation: 
to obtain that
Thus
. (9)
Combined with (8), we just need to estimate the interpolation error bound 
below.
Lemma 5 The solution 
of (1’), (2’) and its piecewise linear interpolant 
satisfy
Proof. According to the splitting of
, we have correspondingly
From Lemma 1 we have
To obtain the estimation for singular component, we use a Taylor expansion
to express the error bound
Continuously, we use the inequality involved a positive monotonically decreasing function g on 
Thus we have
Hence
For the proof of the second statement, we have
thus, lemma 5 follows.
Theorem For
,
defined before, when the Shishkin mesh are applied ,we have the parameter uniform error bound in the energy norm naturally associated with the weak formulation of (1’), (2’)
(10)
Proof. Firstly, we have by triangle inequality and (9)
where in C’s and C1 are stated before. thus we have
Now we use the classical Shishkin mesh (see [11-13]) by setting the mesh transition parameter defined by
and allocate uniformly 
points in each of 
and
. In practice one typically has
, we just acquiesce in this case thus
thus for
,
Also for 
Combining the above two cases reads (10).
Remark. To obtain 
estimation, the standard Aubin-Nitche dual verification skill may be involved.
The superconvergence phenomena on Shishkin mesh for the convection-diffusion problems can be discussed according to Z. Zhang (see [13,14]).
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